# Differential and Integral Calculus

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mathematics, calculus, integral, differential, mir publishers

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Differential and Integral Calculus by N. Piskunov

This text is designed as a course of mathematics for higher technical schools. It contains many worked examples that illustrate the theoretical material and serve as models for solving problems. The first two chapters "Number. Variable. Function" and "Limit. Continuity of a Function" have been made as short as possible. Some of the questions that are usually discussed in these chapters have been put in the third and subsequent chapters without loss of continuity. This has made it possible to take up very early the basic concept of differential calculusâthe derivativeâ which is required in the study of technical subjects. Experience has shown this arrangement of the material to be the best and most convenient for the student.

A large number of problems have been included, many of which illustrate the interrelationships of mathematics and other disciplines. The problems are specially selected (and in sufficient number) for each section of the course thus helping the student to master the theoretical material. To a large extent, this makes the use of a separate book of problems unnecessary and extends the usefulness of this text as a course of mathematics for self-instruction.

The book was translated from the Russian by G. Yankovsky and was published by Mir in 1969. Subsequently it was also republished as a single and two volume format. You can get the book here.

Contents

Preface 11

Chapter I NUMBER. VARIABLE. FUNCTION

1. Real Numbers. Real Numbers as Points on a Number Scale 13

2. The Absolute Value of a Real Number 14

3. Variables and Constants 16

4. The Range of a Variable 16

5. Ordered Variables. Increasing and Decreasing Variables. Bounded Variables 18

6. Function 19

7. Ways of Representing Functions 20

8. Basic Elementary Functions. Elementary Functions 22

9. Algebraic Functions " 26

10. Polar Coordinate System 28

Exercises on Chapter 7 30

Chapter II. LIMIT. CONTINUITY OF A FUNCTION

1. The Limit of a Variable. An Infinitely Large Variable 32

2. The Limit of a Function 35

3. A Function that Approaches Infinity Bounded Functions 38

4. Infinitesimals and Their Basic Properties 42

5. Basic Theorems on Limits 45

6. The Limit of the Function $latex \frac(\sin x)(x) \, x \to 0$. 50

7. The Number e 51

8. Natural Logarithms 56

9. Continuity of Functions 57

10. Certain Properties of Continuous Functions 61

11. Comparing Infinitesimals 63

Exercises on Chapter 77 66

Chapter III. DERIVATIVE AND DIFFERENTIAL

1. Velocity of Motion 69

2. Definition of Derivative 71

3. Geometric Meaning of the Derivative 73

4. Differentiability of Functions 74

5. Finding the Derivatives of Elementary Functions. The Derivative of the Function $y = x^n$ Where n Is Positive and Integral 76

6. Derivatives of the Functions $y=s\nx\ y = cosx$ . 78

7. Derivatives of: a Constant, the Product of a Constant by a Function, a Sum, a Product, and a Quotient 80

8. The Derivative of a Logarithmic Function 84

9. The Derivative of a Composite Function . 85

10. Derivatives of the Functions y = ianx, y = coixt y=\n\x\ 83

11. An Implicit Function and Its Differentiation 89

12. Derivatives of a Power Function for an Arbitrary Real Exponent, of an Exponential Function, and a Composite Exponential Function . . 91

13. An Inverse Function and Its Differentiation 94

14. Inverse Trigonometric Functions and Their Differentiation 98

15. Table of Basic Differentiation Formulas 102

16. Parametric Representation of a Function 103

17. The Equations of Certain Curves in Parametric Form 105

18. The Derivative of a Function Represented Parametrically 108

19. Hyperbolic Functions 110

20. The Differential 113

21. The Geometric Significance of the Differential 117

22. Derivatives of Different Orders 118

23. Differentials of Various Orders 121

24. Different-Order Derivatives of Implicit Functions and of Functions Represented Parametrically 122

25. The Mechanical Significance of the Second Derivative 124

26. The Equations of a Tangent and of a Normal. The Lengths of the Subtangent and the Subnormal 126

27. The Geometric Significance of the Derivative of the Radius Vector with Respect to the Polar Angle 129

Exercises on Chapter III 130

Chapter IV. SOME THEOREMS ON DIFFERENTIABLE FUNCTIONS

1. A Theorem on the Roots of a Derivative (Rolle's Theorem) .... 140

2. A Theorem on Finite Increments (Lagrange's Theorem) 142

3. A Theorem on the Ratio of the Increments of Two Functions (Cauchy's Theorem) - 143

4. The Limit of a Ratio of Two Infinitesimals ( Evaluation of Indeterminate Forms of the Type -rr- J 144

5. The Limit of a Ratio of Two Infinitely Large Quantities (Evaluation 00 \ of Indeterminate Forms of the Type â 147

6. Taylor's Formula 152

7. Expansion of the Functions e*, sin*, and cos x in a Taylor Series . 156

Exercises on Chapter IV 159

Chapter V. INVESTIGATING THE BEHAVIOUR OF FUNCTIONS

1. Statement of the Problem 162

2. Increase and Decrease of a Function 163

3. Maxima and Minima of Functions 164

4. Testing a Differentiable Function for Maximum and Minimum with

a First Derivative 171

5. Testing a Function for Maximum and Minimum with a Second Derivative 174

6. Maxima and Minima of a Function on an Interval 178

7. Applying the Theory of Maxima and Minima of Functions to the Solution of Problems 179

8. Testing a Function for Maximum and Minimum by Means of Taylor's Formula 181

9. Convexity and Concavity of a Curve. Points of Inflection 183

10. Asymptotes ' 189

11. General Plan for Investigating Functions and Constructing Graphs 194

12. Investigating Curves Represented Parametrically 199

Exercises on Chapter V . 203

Chapter VI. THE CURVATURE OF A CURVE

1. The Length of an Arc and Its Derivative 208

2. Curvature 210

3. Calculation of Curvature 212

4. Calculation of the Curvature of a Line Represented Parametrically . . 215

5. Calculation of the Curvature of a Line Given by an Equation of Polar Coordinates 215

6. The Radius and Circle of Curvature. Centre of Curvature. Evolute and Involute 217

7. The Properties of an Evolute 221

8. Approximating the Real Roots of an Equation 225

Exercises on Chapter VI -. . . . 229

Chapter VII. COMPLEX NUMBERS. POLYNOMIALS

1. Complex Numbers. Basic Definitions 233

2. Basic Operations on Complex Numbers : 234

3. Powers and Roots of Complex Numbers 237

4. Exponential Function with Complex Exponent and Its Properties . . 240

5. Euler's Formula. The Exponential Form of a Complex Number . . . 243

6. Factoring a Polynomial . . . . 244

7. The Multiple Roots of a Polynomial 247.

8. Factorisation of a Polynomial in the Case of Complex Roots .... 248

9. Interpolation. Lagrange's Interpolation Formula 250

10. On the Best Approximation of Functions by Polynomials. Chebyshev's Theory . . . . . 252

Exercises on Chapter VII 253

Chapter VIII. FUNCTIONS OF SEVERAL VARIABLES

1. Definition of a Function of Several Variables 255

2. Geometric Representation of a Function of Two Variables ..... 25Â§

3. Partial and Total Increment of a Function 259

4. Continuity of a Function of Several Variables 260

5. Partial Derivatives of a Function of Several Variables 263

6. The Geometric Interpretation of the Partial Derivatives of a Function of Two Variables 264

7. Total Increment and Total Differentials 265

8. Approximation by Total Differentials 268

9. Error Approximation by Differentials 270

10. The Derivative of a Composite Function. The Total Derivative . . . 273

11. The Derivative of a Function Defined Implicitly 276

12. Partial Derivatives of Different Orders 279

13. Level Surfaces .283

14. Directional Derivatives . 284

15. Gradient 286

16. Taylor's Formula for a Function of Two Variables 290

17. Maximum and-Minimum of a Function of Several Variables .... 292

18. Maximum and Minimum of a Function of Several Variables Related

by Given Equations (Conditional Maxima and Minima) 300

19. Singular Points of a Curve 305

Exercises on Chapter VIII 310

Chapter IX. APPLICATIONS OF DIFFERENTIAL CALCULUS TO SOLID GEOMETRY

1. The Equations of a Curve in Space 314

2. The Limit and Derivative of the Vector Function of a Scalar Argument. The Equation of a Tangent to a Curve. The Equation of a Normal Plane 317

3. Rules for Differentiating Vectors (Vector Functions) 322

4. The First and Second Derivatives of a Vector with Respect to the Arc Length. The Curvature of a Curve. The Principal Normal .... 324

5. Osculating Plane. Binormal. Torsion 331

6. A Tangent Plane and Normal to a Surface 336

Exercises on Chapter IX 340

Chapter X, INDEFINITE INTEGRALS

1. Antiderivative and the Indefinite Integral 342

2. Table of Integrals 344

3. Some Properties of an Indefinite Integral 346

4. Integration by Substitution (Change of Variable) 348

5. Integrals of Functions Containing a Quadratic Trinomial 351

6. Integration by Parts 354

7. Rational Fractions. Partial Rational Fractions and Their Integration 357

8. Decomposition of a Rational Fraction into Partial Fractions .... 361

9. Integration of Rational Fractions 365

10. Ostrogradsky's Method 368

11. Integrals of Irrational Functions 371

12. Integrals of the Form [R(x> V'ax2+bx + c)dx . 372

13. Integration of Binomial Differentials 375

14. Integration of Certain Classes of Trigonometric Functions 378

15. Integration of Certain Irrational Functions by Means of Trigonometric Substitutions 383

16. Functions Whose Integrals Cannot Be Expressed in Terms of Elementary Functions 385

Exercises on Chapter X 386

Chapter XI. THE DEFINITE INTEGRAL

1. Statement of the Problem. The Lower and Upper Integral Sums . . . 396

2. The Definite Integral 398

3. Basic Properties o? the Definite Integral . 404

4. Evaluating a Definite Integral. Newton-Leibniz Formula 407

5. Changing the Variable in the Definite Integral 412

6. Integration by Parts 413

7. Improper Integrals 416

8. Approximating Definite Integrals 424

9. Chebyshev's Formula 430

10. Integrals Dependent on a Parameter 435

Exercises on Chapter XI 438

Chapter XII. GEOMETRIC AND MECHANICAL APPLICATIONS OF THE DEFINITE INTEGRAL

1. Computing Areas in Rectangular Coordinates 442

2. The Area of a Curvilinear Sector in Polar Coordinates ' . 445

3. The Arc Length of a Curve . 447

4. Computing the Volume of a Solid from the Areas of Parallel Sections (Volumes by Slicing) 453

5. The Volume of a Solid of Revolution 455

6. The Surface of a Solid of Revolution 455

7. Computing Work by the Definite Integral 457

8. Coordinates of the Centre of Gravity 459

Exercises on Chapter XII 462

Chapter XIII. DIFFERENTIAL EQUATIONS

1. Statement of the Problem. The Equation of Motion of a Body with Resistance of the Medium Proportional to the Velocity. The Equation of a Catenary 469

2. Definitions \ . . , 472

3. First-Order Differential Equations (General Notions) . 473

4. Equations with Separated and Separable Variables. The Problem of the Disintegration of Radium 478

5. Homogeneous First-Order Equations 482

6. Equations Reducible to Homogeneous Equations 484

7. First-Order Linear Equations 487

8. Bernoulli's Equation 490

9. Exact Differential Equations 492

10. Integrating Factor 495

11. The Envelope of a Family of Curves 497

12. Singular Solutions of a First-Order Differential Equation 504

13. Clairaut's Equation 505

14. Lagrange's Equation 507

15. Orthogonal and Isogonal Trajectories 509

16. Higher-Order Differential Equations (Fundamentals) 514

17. An Equation of the Form y = f(x) 516

18. Some Types of Second-Order Differential Equations Reducible to First-Order Equations . . 518

19. Graphical Method of Integrating Second-Order Differential Equations 527

20. Homogeneous Linear Equations. Definitions and General Properties 528

21. Second-Order Homogeneous Linear Equations with Constant Coefficients ( 535

22. Homogeneous Linear Equations of the Aith Order with Constant Coefficients 539

23. Nonhomogeneous Second-Order Linear Equations 541

24. Nonhomogeneous Second-Order Linear Equations with Constant Coefficients 545

25. Higher-Order Nonhomogeneous Linear Equations 551

26. The Differential -Equation of Mechanical Vibrations 555

27. Free Oscillations' 557

28. Forced Oscillations 559

29. Systems of Ordinary Differential Equations 563

30. Systems of Linear Differential Equations with Constant Coefficients 569

31. On Lyapunov's Theory of Stability 576

32. Euler's Method of Approximate Solution of First-Order Differential Equations . 581

33. A Difference Method for Approximate Solution of Differential Equations Based on Taylor's Formula. Adams Method 584

34. An Approximate Method for Integrating Systems of First-Order Differential Equations 591

Exercises on Chapter XIII 595

Chapter XIV. MULTIPLE INTEGRALS

1. Double Integrals . 608

2. Calculating Double Integrals 610

3. Calculating Double Integrals (Continued) 617

4. Calculating Areas and Volumes by Means of Double Integrals .... 623

5. The Double Integral in Polar Coordinates 626

6. Changing Variables in a Double Integral (General Case) 633

7. Computing the Area of a Surface 638

8. The Density of Distribution of Matter and the Double Integral . . . 642

9. The Moment of Inertia of the Area of a Plane Figure 643

10. The Coordinates of the Centre of Gravity of the Area of a Plane Figure 648

11. Triple Integrals 650

12. Evaluating a Triple Integral 651

13. Change of Variables in a Triple Integral 656

14. The Moment of Inertia and the Coordinates of the Centre of Gravity of a Solid . 660

15. Computing Integrals Dependent on a Parameter . 662

Exercises on Chapter XIV . . . 663

Chapter XV. LINE INTEGRALS AND SURFACE INTEGRALS

1. Line Integrals 670

2. Evaluating a Line Integral 673

3. Green's Formula 679

4. Conditions for a Line Integral Being Independent of the Path of Integration 681

5. Surface Integrals . 687

6. Evaluating Surface Integrals 689

7. Stokes' Formula 692

8. Ostrogradsky's Formula 697

9. The Hamiltonian Operator and Certain Applications of It 700

Exercises on Chapter XV 703

Chapter XVI. SERIES

1. Series. Sum of a Series 710

2. Necessary Condition for Convergence of a Series 713

3. Comparing Series with Positive Terms 716

4. D'Alembert's Test 718

5. Cauchy's Test 721

6. The Integral Test for Convergence of a Series .723

7. Alternating Series. Leibniz' Theorem 727

8. PIus-and-Minus Series. Absolute and Contitional Convergence .... 729

9. Functional Series 733

10. Majorised Series 734

11. The Continuity of the Sum of a Series 736

12. Integration and Differentiation of Series 739

13. Power Series. Interval of Convergence 742

14. Differentiation of Power Series 747

15. Series in Powers of â * âa 748

16. Taylor's Series and Maclaurin's Series .' 750

17. Examples of Expansion of Functions in Series 751

18. Euler's Formula 753

19. The Binomial Series 754

20. Expansion of the Function In (\+x) in a Power Series. Computing Logarithms 756

21. Integration by Use of Series (Calculating Definite Integrals) .... 758

22. Integrating Differential Equations by Means of Series ....... 760

23. Bessel's Equation 763

Exercises on Chapter XVI 768

Chapter XVII. FOURIER SERIES

1. Definition. Statement of the Problem 776

2. Expansions of Functions in Fourier Series 780

3. A Remark on the Expansion of a Periodic Function in a Fourier Series 785

4. Fourier Series for Even and Odd Functions 787

5. The Fourier Series for a Function with Period 2/ 789

6. On the Expansion of a Nonperiodic Function in a Fourier Series . . 791

7. Approximation by a Trigonometric Polynomial of a Function Represented in the Mean 792

8. The Dirichlet Integral 798

9. The Convergence of a Fourier Series at a Given Point . . . . . . .801

10. Certain Sufficient Conditions for the Convergence of a Fourier Series 802

11. Practical Harmonic Analysis 805

12. Fourier Integral 810

13. The Fourier Integral in Complex Form 810

Exercises on Chapter XVII 812

Chapter XVIII. EQUATIONS OF MATHEMATICAL PHYSICS

1. Basic Types of Equations of Mathematical Physics . 815

2. Derivation of the Equation of Oscillations of a String. Formulation of the Boundary-Value Problem. Derivation of Equations of Electric Oscillations in Wires : 816

3. Solution of the Equation of Oscillations of a String by the Method

of Separation of Variables (The Fourier Method) 820

4. The Equation for Propagation of Heat in a Rod. Formulation of the Boundary-Value Problem 823

5. Heat Propagation in Space 825

6. Solution of the First Boundary-Value Problem for the Heat- Conductivity Equation by the Method of Finite Differences 829

7. Propagation of Heat in an Unbounded Rod 831

8. Problems That Reduce to Investigating Solutions of the Laplace Equation. Stating Boundary-Value Problems 836

9. The Laplace Equation in Cylindrical Coordinates. Solution of the Dirichlet Problem for a Ring with Constant Values of the Desired Function on the Inner and Outer Circumferences 841

10. The Solution of Dirichlet's Problem for a Circle 843

11. Solution of the Dirichlet Problem by the Method of Finite Differences 847

Exercises on Chapter XVIII 850

Chapter XIX. OPERATIONAL CALCULUS AND CERTAIN OF ITS APPLICATIONS

1. The Initial Function and Its Transform 854

2. Transforms of the Functions.oQ(t), sin/, cost 855

3. The Transform of a Function with Changed Scale of the Independent Variable. Transforms of the Functions sin at, cos at 856

4. The Linearity Property of a Transform 857

5. The Shift Theorem r 858

6. Transforms of the Functionse~at, sinh at, cosh at, e-aisinatt e~a/ cos at 858

7. Differentiation of Transforms 860

8. The Transforms of Derivatives 861

9. Table of Transforms ~. 862

10. An Auxiliary Equation for a Given Differential Equation 864

11. Decomposition Theorem . 867

12. Examples of Solutions of Differential Equations and Systems of Differential Equations by the Operational /Method 869

13. The Convolution Theorem 871

14. The Differential Equations of Mechanical Oscillations. The Differential Equations of Electric-Circuit Theory 873

15. Solution of the Differential Oscillation Equation 874

16. Investigating Free Oscillations 875

17. Investigating Mechanical and Electrical Oscillations in the Case of a Periodic External Force 876

18. Solving the Oscillation Equation in the Case of Resonance . . . . 878

19. The Delay Theorem 879

Exercises on Chapter XIX 880

Subject Index

This text is designed as a course of mathematics for higher technical schools. It contains many worked examples that illustrate the theoretical material and serve as models for solving problems. The first two chapters "Number. Variable. Function" and "Limit. Continuity of a Function" have been made as short as possible. Some of the questions that are usually discussed in these chapters have been put in the third and subsequent chapters without loss of continuity. This has made it possible to take up very early the basic concept of differential calculusâthe derivativeâ which is required in the study of technical subjects. Experience has shown this arrangement of the material to be the best and most convenient for the student.

A large number of problems have been included, many of which illustrate the interrelationships of mathematics and other disciplines. The problems are specially selected (and in sufficient number) for each section of the course thus helping the student to master the theoretical material. To a large extent, this makes the use of a separate book of problems unnecessary and extends the usefulness of this text as a course of mathematics for self-instruction.

The book was translated from the Russian by G. Yankovsky and was published by Mir in 1969. Subsequently it was also republished as a single and two volume format. You can get the book here.

Contents

Preface 11

Chapter I NUMBER. VARIABLE. FUNCTION

1. Real Numbers. Real Numbers as Points on a Number Scale 13

2. The Absolute Value of a Real Number 14

3. Variables and Constants 16

4. The Range of a Variable 16

5. Ordered Variables. Increasing and Decreasing Variables. Bounded Variables 18

6. Function 19

7. Ways of Representing Functions 20

8. Basic Elementary Functions. Elementary Functions 22

9. Algebraic Functions " 26

10. Polar Coordinate System 28

Exercises on Chapter 7 30

Chapter II. LIMIT. CONTINUITY OF A FUNCTION

1. The Limit of a Variable. An Infinitely Large Variable 32

2. The Limit of a Function 35

3. A Function that Approaches Infinity Bounded Functions 38

4. Infinitesimals and Their Basic Properties 42

5. Basic Theorems on Limits 45

6. The Limit of the Function $latex \frac(\sin x)(x) \, x \to 0$. 50

7. The Number e 51

8. Natural Logarithms 56

9. Continuity of Functions 57

10. Certain Properties of Continuous Functions 61

11. Comparing Infinitesimals 63

Exercises on Chapter 77 66

Chapter III. DERIVATIVE AND DIFFERENTIAL

1. Velocity of Motion 69

2. Definition of Derivative 71

3. Geometric Meaning of the Derivative 73

4. Differentiability of Functions 74

5. Finding the Derivatives of Elementary Functions. The Derivative of the Function $y = x^n$ Where n Is Positive and Integral 76

6. Derivatives of the Functions $y=s\nx\ y = cosx$ . 78

7. Derivatives of: a Constant, the Product of a Constant by a Function, a Sum, a Product, and a Quotient 80

8. The Derivative of a Logarithmic Function 84

9. The Derivative of a Composite Function . 85

10. Derivatives of the Functions y = ianx, y = coixt y=\n\x\ 83

11. An Implicit Function and Its Differentiation 89

12. Derivatives of a Power Function for an Arbitrary Real Exponent, of an Exponential Function, and a Composite Exponential Function . . 91

13. An Inverse Function and Its Differentiation 94

14. Inverse Trigonometric Functions and Their Differentiation 98

15. Table of Basic Differentiation Formulas 102

16. Parametric Representation of a Function 103

17. The Equations of Certain Curves in Parametric Form 105

18. The Derivative of a Function Represented Parametrically 108

19. Hyperbolic Functions 110

20. The Differential 113

21. The Geometric Significance of the Differential 117

22. Derivatives of Different Orders 118

23. Differentials of Various Orders 121

24. Different-Order Derivatives of Implicit Functions and of Functions Represented Parametrically 122

25. The Mechanical Significance of the Second Derivative 124

26. The Equations of a Tangent and of a Normal. The Lengths of the Subtangent and the Subnormal 126

27. The Geometric Significance of the Derivative of the Radius Vector with Respect to the Polar Angle 129

Exercises on Chapter III 130

Chapter IV. SOME THEOREMS ON DIFFERENTIABLE FUNCTIONS

1. A Theorem on the Roots of a Derivative (Rolle's Theorem) .... 140

2. A Theorem on Finite Increments (Lagrange's Theorem) 142

3. A Theorem on the Ratio of the Increments of Two Functions (Cauchy's Theorem) - 143

4. The Limit of a Ratio of Two Infinitesimals ( Evaluation of Indeterminate Forms of the Type -rr- J 144

5. The Limit of a Ratio of Two Infinitely Large Quantities (Evaluation 00 \ of Indeterminate Forms of the Type â 147

6. Taylor's Formula 152

7. Expansion of the Functions e*, sin*, and cos x in a Taylor Series . 156

Exercises on Chapter IV 159

Chapter V. INVESTIGATING THE BEHAVIOUR OF FUNCTIONS

1. Statement of the Problem 162

2. Increase and Decrease of a Function 163

3. Maxima and Minima of Functions 164

4. Testing a Differentiable Function for Maximum and Minimum with

a First Derivative 171

5. Testing a Function for Maximum and Minimum with a Second Derivative 174

6. Maxima and Minima of a Function on an Interval 178

7. Applying the Theory of Maxima and Minima of Functions to the Solution of Problems 179

8. Testing a Function for Maximum and Minimum by Means of Taylor's Formula 181

9. Convexity and Concavity of a Curve. Points of Inflection 183

10. Asymptotes ' 189

11. General Plan for Investigating Functions and Constructing Graphs 194

12. Investigating Curves Represented Parametrically 199

Exercises on Chapter V . 203

Chapter VI. THE CURVATURE OF A CURVE

1. The Length of an Arc and Its Derivative 208

2. Curvature 210

3. Calculation of Curvature 212

4. Calculation of the Curvature of a Line Represented Parametrically . . 215

5. Calculation of the Curvature of a Line Given by an Equation of Polar Coordinates 215

6. The Radius and Circle of Curvature. Centre of Curvature. Evolute and Involute 217

7. The Properties of an Evolute 221

8. Approximating the Real Roots of an Equation 225

Exercises on Chapter VI -. . . . 229

Chapter VII. COMPLEX NUMBERS. POLYNOMIALS

1. Complex Numbers. Basic Definitions 233

2. Basic Operations on Complex Numbers : 234

3. Powers and Roots of Complex Numbers 237

4. Exponential Function with Complex Exponent and Its Properties . . 240

5. Euler's Formula. The Exponential Form of a Complex Number . . . 243

6. Factoring a Polynomial . . . . 244

7. The Multiple Roots of a Polynomial 247.

8. Factorisation of a Polynomial in the Case of Complex Roots .... 248

9. Interpolation. Lagrange's Interpolation Formula 250

10. On the Best Approximation of Functions by Polynomials. Chebyshev's Theory . . . . . 252

Exercises on Chapter VII 253

Chapter VIII. FUNCTIONS OF SEVERAL VARIABLES

1. Definition of a Function of Several Variables 255

2. Geometric Representation of a Function of Two Variables ..... 25Â§

3. Partial and Total Increment of a Function 259

4. Continuity of a Function of Several Variables 260

5. Partial Derivatives of a Function of Several Variables 263

6. The Geometric Interpretation of the Partial Derivatives of a Function of Two Variables 264

7. Total Increment and Total Differentials 265

8. Approximation by Total Differentials 268

9. Error Approximation by Differentials 270

10. The Derivative of a Composite Function. The Total Derivative . . . 273

11. The Derivative of a Function Defined Implicitly 276

12. Partial Derivatives of Different Orders 279

13. Level Surfaces .283

14. Directional Derivatives . 284

15. Gradient 286

16. Taylor's Formula for a Function of Two Variables 290

17. Maximum and-Minimum of a Function of Several Variables .... 292

18. Maximum and Minimum of a Function of Several Variables Related

by Given Equations (Conditional Maxima and Minima) 300

19. Singular Points of a Curve 305

Exercises on Chapter VIII 310

Chapter IX. APPLICATIONS OF DIFFERENTIAL CALCULUS TO SOLID GEOMETRY

1. The Equations of a Curve in Space 314

2. The Limit and Derivative of the Vector Function of a Scalar Argument. The Equation of a Tangent to a Curve. The Equation of a Normal Plane 317

3. Rules for Differentiating Vectors (Vector Functions) 322

4. The First and Second Derivatives of a Vector with Respect to the Arc Length. The Curvature of a Curve. The Principal Normal .... 324

5. Osculating Plane. Binormal. Torsion 331

6. A Tangent Plane and Normal to a Surface 336

Exercises on Chapter IX 340

Chapter X, INDEFINITE INTEGRALS

1. Antiderivative and the Indefinite Integral 342

2. Table of Integrals 344

3. Some Properties of an Indefinite Integral 346

4. Integration by Substitution (Change of Variable) 348

5. Integrals of Functions Containing a Quadratic Trinomial 351

6. Integration by Parts 354

7. Rational Fractions. Partial Rational Fractions and Their Integration 357

8. Decomposition of a Rational Fraction into Partial Fractions .... 361

9. Integration of Rational Fractions 365

10. Ostrogradsky's Method 368

11. Integrals of Irrational Functions 371

12. Integrals of the Form [R(x> V'ax2+bx + c)dx . 372

13. Integration of Binomial Differentials 375

14. Integration of Certain Classes of Trigonometric Functions 378

15. Integration of Certain Irrational Functions by Means of Trigonometric Substitutions 383

16. Functions Whose Integrals Cannot Be Expressed in Terms of Elementary Functions 385

Exercises on Chapter X 386

Chapter XI. THE DEFINITE INTEGRAL

1. Statement of the Problem. The Lower and Upper Integral Sums . . . 396

2. The Definite Integral 398

3. Basic Properties o? the Definite Integral . 404

4. Evaluating a Definite Integral. Newton-Leibniz Formula 407

5. Changing the Variable in the Definite Integral 412

6. Integration by Parts 413

7. Improper Integrals 416

8. Approximating Definite Integrals 424

9. Chebyshev's Formula 430

10. Integrals Dependent on a Parameter 435

Exercises on Chapter XI 438

Chapter XII. GEOMETRIC AND MECHANICAL APPLICATIONS OF THE DEFINITE INTEGRAL

1. Computing Areas in Rectangular Coordinates 442

2. The Area of a Curvilinear Sector in Polar Coordinates ' . 445

3. The Arc Length of a Curve . 447

4. Computing the Volume of a Solid from the Areas of Parallel Sections (Volumes by Slicing) 453

5. The Volume of a Solid of Revolution 455

6. The Surface of a Solid of Revolution 455

7. Computing Work by the Definite Integral 457

8. Coordinates of the Centre of Gravity 459

Exercises on Chapter XII 462

Chapter XIII. DIFFERENTIAL EQUATIONS

1. Statement of the Problem. The Equation of Motion of a Body with Resistance of the Medium Proportional to the Velocity. The Equation of a Catenary 469

2. Definitions \ . . , 472

3. First-Order Differential Equations (General Notions) . 473

4. Equations with Separated and Separable Variables. The Problem of the Disintegration of Radium 478

5. Homogeneous First-Order Equations 482

6. Equations Reducible to Homogeneous Equations 484

7. First-Order Linear Equations 487

8. Bernoulli's Equation 490

9. Exact Differential Equations 492

10. Integrating Factor 495

11. The Envelope of a Family of Curves 497

12. Singular Solutions of a First-Order Differential Equation 504

13. Clairaut's Equation 505

14. Lagrange's Equation 507

15. Orthogonal and Isogonal Trajectories 509

16. Higher-Order Differential Equations (Fundamentals) 514

17. An Equation of the Form y = f(x) 516

18. Some Types of Second-Order Differential Equations Reducible to First-Order Equations . . 518

19. Graphical Method of Integrating Second-Order Differential Equations 527

20. Homogeneous Linear Equations. Definitions and General Properties 528

21. Second-Order Homogeneous Linear Equations with Constant Coefficients ( 535

22. Homogeneous Linear Equations of the Aith Order with Constant Coefficients 539

23. Nonhomogeneous Second-Order Linear Equations 541

24. Nonhomogeneous Second-Order Linear Equations with Constant Coefficients 545

25. Higher-Order Nonhomogeneous Linear Equations 551

26. The Differential -Equation of Mechanical Vibrations 555

27. Free Oscillations' 557

28. Forced Oscillations 559

29. Systems of Ordinary Differential Equations 563

30. Systems of Linear Differential Equations with Constant Coefficients 569

31. On Lyapunov's Theory of Stability 576

32. Euler's Method of Approximate Solution of First-Order Differential Equations . 581

33. A Difference Method for Approximate Solution of Differential Equations Based on Taylor's Formula. Adams Method 584

34. An Approximate Method for Integrating Systems of First-Order Differential Equations 591

Exercises on Chapter XIII 595

Chapter XIV. MULTIPLE INTEGRALS

1. Double Integrals . 608

2. Calculating Double Integrals 610

3. Calculating Double Integrals (Continued) 617

4. Calculating Areas and Volumes by Means of Double Integrals .... 623

5. The Double Integral in Polar Coordinates 626

6. Changing Variables in a Double Integral (General Case) 633

7. Computing the Area of a Surface 638

8. The Density of Distribution of Matter and the Double Integral . . . 642

9. The Moment of Inertia of the Area of a Plane Figure 643

10. The Coordinates of the Centre of Gravity of the Area of a Plane Figure 648

11. Triple Integrals 650

12. Evaluating a Triple Integral 651

13. Change of Variables in a Triple Integral 656

14. The Moment of Inertia and the Coordinates of the Centre of Gravity of a Solid . 660

15. Computing Integrals Dependent on a Parameter . 662

Exercises on Chapter XIV . . . 663

Chapter XV. LINE INTEGRALS AND SURFACE INTEGRALS

1. Line Integrals 670

2. Evaluating a Line Integral 673

3. Green's Formula 679

4. Conditions for a Line Integral Being Independent of the Path of Integration 681

5. Surface Integrals . 687

6. Evaluating Surface Integrals 689

7. Stokes' Formula 692

8. Ostrogradsky's Formula 697

9. The Hamiltonian Operator and Certain Applications of It 700

Exercises on Chapter XV 703

Chapter XVI. SERIES

1. Series. Sum of a Series 710

2. Necessary Condition for Convergence of a Series 713

3. Comparing Series with Positive Terms 716

4. D'Alembert's Test 718

5. Cauchy's Test 721

6. The Integral Test for Convergence of a Series .723

7. Alternating Series. Leibniz' Theorem 727

8. PIus-and-Minus Series. Absolute and Contitional Convergence .... 729

9. Functional Series 733

10. Majorised Series 734

11. The Continuity of the Sum of a Series 736

12. Integration and Differentiation of Series 739

13. Power Series. Interval of Convergence 742

14. Differentiation of Power Series 747

15. Series in Powers of â * âa 748

16. Taylor's Series and Maclaurin's Series .' 750

17. Examples of Expansion of Functions in Series 751

18. Euler's Formula 753

19. The Binomial Series 754

20. Expansion of the Function In (\+x) in a Power Series. Computing Logarithms 756

21. Integration by Use of Series (Calculating Definite Integrals) .... 758

22. Integrating Differential Equations by Means of Series ....... 760

23. Bessel's Equation 763

Exercises on Chapter XVI 768

Chapter XVII. FOURIER SERIES

1. Definition. Statement of the Problem 776

2. Expansions of Functions in Fourier Series 780

3. A Remark on the Expansion of a Periodic Function in a Fourier Series 785

4. Fourier Series for Even and Odd Functions 787

5. The Fourier Series for a Function with Period 2/ 789

6. On the Expansion of a Nonperiodic Function in a Fourier Series . . 791

7. Approximation by a Trigonometric Polynomial of a Function Represented in the Mean 792

8. The Dirichlet Integral 798

9. The Convergence of a Fourier Series at a Given Point . . . . . . .801

10. Certain Sufficient Conditions for the Convergence of a Fourier Series 802

11. Practical Harmonic Analysis 805

12. Fourier Integral 810

13. The Fourier Integral in Complex Form 810

Exercises on Chapter XVII 812

Chapter XVIII. EQUATIONS OF MATHEMATICAL PHYSICS

1. Basic Types of Equations of Mathematical Physics . 815

2. Derivation of the Equation of Oscillations of a String. Formulation of the Boundary-Value Problem. Derivation of Equations of Electric Oscillations in Wires : 816

3. Solution of the Equation of Oscillations of a String by the Method

of Separation of Variables (The Fourier Method) 820

4. The Equation for Propagation of Heat in a Rod. Formulation of the Boundary-Value Problem 823

5. Heat Propagation in Space 825

6. Solution of the First Boundary-Value Problem for the Heat- Conductivity Equation by the Method of Finite Differences 829

7. Propagation of Heat in an Unbounded Rod 831

8. Problems That Reduce to Investigating Solutions of the Laplace Equation. Stating Boundary-Value Problems 836

9. The Laplace Equation in Cylindrical Coordinates. Solution of the Dirichlet Problem for a Ring with Constant Values of the Desired Function on the Inner and Outer Circumferences 841

10. The Solution of Dirichlet's Problem for a Circle 843

11. Solution of the Dirichlet Problem by the Method of Finite Differences 847

Exercises on Chapter XVIII 850

Chapter XIX. OPERATIONAL CALCULUS AND CERTAIN OF ITS APPLICATIONS

1. The Initial Function and Its Transform 854

2. Transforms of the Functions.oQ(t), sin/, cost 855

3. The Transform of a Function with Changed Scale of the Independent Variable. Transforms of the Functions sin at, cos at 856

4. The Linearity Property of a Transform 857

5. The Shift Theorem r 858

6. Transforms of the Functionse~at, sinh at, cosh at, e-aisinatt e~a/ cos at 858

7. Differentiation of Transforms 860

8. The Transforms of Derivatives 861

9. Table of Transforms ~. 862

10. An Auxiliary Equation for a Given Differential Equation 864

11. Decomposition Theorem . 867

12. Examples of Solutions of Differential Equations and Systems of Differential Equations by the Operational /Method 869

13. The Convolution Theorem 871

14. The Differential Equations of Mechanical Oscillations. The Differential Equations of Electric-Circuit Theory 873

15. Solution of the Differential Oscillation Equation 874

16. Investigating Free Oscillations 875

17. Investigating Mechanical and Electrical Oscillations in the Case of a Periodic External Force 876

18. Solving the Oscillation Equation in the Case of Resonance . . . . 878

19. The Delay Theorem 879

Exercises on Chapter XIX 880

Subject Index

Identifier
DifferentialAndIntegralCalculus_109

Identifier-ark
ark:/13960/t6d235f2v

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Reviews

**Reviewer:**InderS - favoritefavoritefavoritefavorite - May 30, 2014

**Subject:**Differential and Integral Calculus

Really good book with lots of examples and well-explained theory